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A generalized antithetic time series theory for exponentially derived antithetic random variables is developed. The correlation function between a generalized gamma distributed random variable and its
*p*th exponent is derived. We prove that the correlation approaches minus one as the exponent approaches zero from the left and the shape parameter approaches infinity.

Serially correlated random variables arise in ways that both benefit and bias mathematical models for science, engineering and economics. In one widespread example, a mathematical formula is used to create uniformly distributed pseudo random numbers for use in Monte Carlo simulation. The numbers are serially correlated because they are generated by a formula. The benefit is that the same pseudo random numbers can be recreated at will, and two or more simulation experiments can be compared without regard to the pseudo random numbers. The correlation is designed to be very small so as not to bias the results of a simulation experiment. Still, some bias is unavoidable when using serially correlated numbers (Ferrenberg, Lanau, and Wong [

Another wide spread example is a regression model in which the dependent variable is serially correlated. The result is biased model parameter estimates because the independence assumption of the Gauss-Markov theorem is violated (see Griliches [

Economics researchers have commented on serial correlation bias. Hendry and Mizon [

Inversely correlated random numbers were suggested by Hammersley and Morton [

The theory of combining antithetic lognormally distributed random variables that contain negatively cor- related components was introduced by Ridley [

then if the correlation between

bined so as to eliminate bias in fitted values associated with any autoregressive time series model (see the Ridley [

The implication for using a biased mathematical model to investigate economic, engineering and scientific phenomena is that estimates obtained from the model are biased. Estimates of future values extrapolated from the model are also biased. As the forecast horizon increases, the bias accumulates and the extrapolations diverge from the actual values. This is most pronounced in the case of investigations into global warming phenomena. There, the horizon is by definition very far into the future. The smallest bias will accumulate, so much so that conclusions may be as much an artifact of the mathematical model as they are about climate dynamics. Com- bining antithetic extrapolations can dynamically remove the bias in the extrapolated values.

The antithetic gamma variables discussed in this research are defined as follows.

Definition 1. Two random variables are antithetic if their correlation is negative. A bivariate collection of random variables is asymptotically antithetic if its limiting correlation approaches minus one asymptotically (see antithetic gamma variables theorem below).

Definition 2.

In this paper, we extend the discovery by Ridley [

One purpose of this paper is to derive an analytical function for the correlation between X and

The paper is organized as follows. In Section 2 we review the gamma distribution. In Section 3 we derive the analytic function for the correlation. In Section 4 we prove its limiting value. In Section 5 we use MATLAB [

The gamma distribution is very important for technical reasons, since it is the parent of the exponential distribution and can explain many other distributions. Its probability distribution function (pdf) (see Hogg and Ledolter [

where

A graph of the gamma probability density function for

Let

Therefore, since

the second moment is

and the variance is

Let

and

Using Equation (3)

Therefore, using Equations (3) and (6), Equation (5) becomes

Since

From Equation (4)

or

The gamma function

be satisfied when

Theorem 1. If

See proof in Appendix B.

The effect of p on the correlation is demonstrated by calculating the correlation coefficient from Equation (8) for various values of

that are more symmetrical. Also, as

To validate Equation (8), the MATLAB [

The results are shown in

Consider an autoregressive time series

tively correlated. The antithetic component

where the exponent of the power transformation is set to the small negative value

A shift parameter

antithetic time series is rewritten and computed from

Of the terms p,

To illustrate, consider a model fitted to computer simulated data based on stationary autoregressive processes, containing 1060 observations generated from

The results are shown in

The correlation between a gamma distributed random variable and its pth power was derived. It was proved that the correlation approaches minus one as p approaches zero from the left and the shape parameter approaches infinity. This counterintuitive result extends a previous finding of the similar result for lognormally distributed random variables. The gamma distribution was modified so as to emulate a range of distributions, showing that

Fitted mse | |||||
---|---|---|---|---|---|

Original | Combined | Reduction % | |||

5 | 370 | −46 | 1.377 | 1.209 | 12.2 |

10 | 520 | −39 | 4.753 | 4.394 | 7.6 |

15 | 481 | −17 | 6.892 | 6.732 | 2.3 |

20 | 575 | −11 | 6.922 | 6.824 | 1.4 |

25 | 529 | −43 | 9.686 | 9.304 | 3.9 |

Average | 5.5 |

Forecast origin n | Fitted mse | Forecast mse | ||||
---|---|---|---|---|---|---|

Original | Combined | Original | Combined | |||

50 | 370 | −46 | 1.377 | 1.209 | 6.158 | 5.760 |

51 | 411 | −46 | 1.351 | 1.185 | 6.230 | 5.407 |

52 | 252 | −46 | 1.354 | 1.202 | 6.493 | 5.641 |

53 | 446 | −39 | 1.397 | 1.266 | 6.071 | 5.943 |

54 | 310 | −55 | 1.371 | 1.160 | 6.070 | 5.611 |

55 | 261 | −37 | 1.356 | 1.254 | 5.941 | 5.549 |

56 | 465 | −41 | 1.345 | 1.189 | 6.108 | 5.720 |

57 | 400 | −39 | 1.331 | 1.218 | 6.234 | 5.592 |

58 | 507 | −36 | 1.312 | 1.191 | 6.142 | 6.224 |

59 | 405 | −32 | 1.290 | 1.175 | 6.044 | 5.687 |

60 | 354 | −62 | 1.434 | 1.207 | 5.600 | 5.300 |

Average | 1.356 | 1.205 | 6.099 | 5.676 | ||

Combined reduction % | 11.1 | 6.9 |

Forecast horizon N | Forecast mse | ||
---|---|---|---|

Original | Combined | Reduction% | |

100 | 5.973 | 5.581 | 6.6 |

150 | 6.529 | 6.059 | 7.2 |

200 | 5.426 | 5.220 | 3.8 |

250 | 6.559 | 6.152 | 6.2 |

300 | 6.455 | 6.181 | 4.2 |

350 | 6.390 | 6.051 | 5.3 |

400 | 5.982 | 5.630 | 5.9 |

450 | 5.906 | 5.554 | 6.0 |

500 | 5.972 | 5.607 | 6.1 |

550 | 6.114 | 5.699 | 6.8 |

600 | 6.063 | 5.616 | 7.4 |

650 | 5.818 | 5.401 | 7.2 |

700 | 5.632 | 5.243 | 6.9 |

Average | 6.063 | 5.692 | 6.1 |

antithetic time series analysis can be generalized to all data distributions that are likely to occur in practice. The gamma distribution is unimodal. A suggestion for future research is to investigate the correlation between a random variable and its pth power when its distribution is multimodal. Another suggestion is to compare the effectiveness of the Hammersley and Morton [

The authors would like to thank Dennis Duke for probing questions and good discussions.

PierreNgnepieba,DennisRidley,11, (2015) General Theory of Antithetic Time Series. Journal of Applied Mathematics and Physics,03,1726-1741. doi: 10.4236/jamp.2015.312197

The pth moment of the gamma distribution is derived as follows:

Multiplying and dividing by

Since

and equation (A.2) becomes

By applying the Taylor expansion around

where

The combination of equations (B.1)-(B.3) reduces Equation (8) to

Therefore,

By using the polygamma function (see Abramowitz and Stegun [

Equation (B.4) is transformed into

The digamma function for real

(see also Bernado [

Its derivative is the polygamma function

And,

From which,

Consider a gamma distributed time series

where

As p approached zero from the left, near perfect correlation between

Now, suppose that

is a time series model. If there is any bias due either to serial correlation in

To remove this bias, we power transform

where

Denoting sample standard deviation by s and correlation coefficient by

(see also the Ridley [

Both estimates

where

Consider the error in

respect to

The steps for obtaining the combined antithetic fitted values are outlined as follows:

Step 1: Estimate the model parameters and fitted values

Step 2: Set

Step 3: Calculate

Step 4: Calculate

Likewise, the unbiased combined estimate of a future value at time

Consider a gamma distributed time series

Therefore, as

(see also Fuller [

where

due only to errors resulting from serial correlation. Therefore,

Next, consider another fitted value

Substituting for

where

is the antithetic error due to the serial correlation, but corresponding to

The expansion of

contain the constant

Now

Substituting from Equation (D.2) and (D.6) and since

Substituting for

Substituting from (D.3) and factoring out

and since

ways in which the combined error variance can be less than the original error variance in Equation (D.3). In

particular when

vanishes. The only error variance remaining will be due purely to random error unexplained by the original model.